What this calculator does
Chebyshev theorem guarantees that at least 1 − 1/k² of any data set lies within k standard deviations of the mean. At k = 2 that is 75%, and it holds for every distribution without exception.
The contrast with the empirical rule is the point. For normal data, two standard deviations captures about 95%. Chebyshev promises only 75%, but it promises it for skewed data, bimodal data, data with heavy tails, anything at all. It buys universality at the cost of being loose, which is exactly the right trade when the distribution is unknown.
The formula
The minimum share is 1 minus 1 over k squared, valid for any k greater than 1. Below k = 1 the bound gives nothing, since it would be negative or zero. The theorem is a worst case guarantee rather than an estimate: real data almost always has more inside the interval than the bound requires, often far more.
| Term | Meaning |
|---|---|
| Chebyshev bound | At least 1 − 1/k² of the data within k standard deviations. |
| Distribution-free | Requiring no assumption about the shape of the data. |
| Worst case | The bound is attained only by specially constructed distributions; ordinary data beats it comfortably. |
| k | The number of standard deviations. The bound is useful only for k greater than 1. |
The inputs explained
| Field | What to enter |
|---|---|
| Number of standard deviations (k) | Number of standard deviations. Values at or below 1 give no useful bound. |
| Mean (optional, for the interval) | Optional, used only to report the interval in the original units. |
| Standard deviation (optional, for the interval) | Optional, used only to report the interval. |
When to use it
Working with unknown distributions
When there is no basis for assuming normality, Chebyshev gives the only guarantee available.
Setting a conservative bound
A guarantee that cannot fail is worth more than a tighter estimate that might, particularly in risk and safety work.
Teaching why normality matters
Comparing 75% against 95% at two standard deviations shows exactly what the normal assumption buys.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
What share is guaranteed at each k?
The guaranteed minimum share at a range of k values.
| Standard deviations (k) | Minimum share within k SDs | Maximum share beyond k SDs | Interval (mean ± k·SD) |
|---|---|---|---|
| k = 1.5 | 55.6% | 44.4% | 77.50 to 122.50 |
| k = 2 | 75.0% | 25.0% | 70.00 to 130.00 |
| k = 2.5 | 84.0% | 16.0% | 62.50 to 137.50 |
| k = 3 | 88.9% | 11.1% | 55.00 to 145.00 |
| k = 4 | 93.8% | 6.25% | 40.00 to 160.00 |
Questions
What is the difference between Chebyshev and the empirical rule?
The empirical rule describes normal data specifically and gives tight figures: 68, 95 and 99.7%. Chebyshev gives a minimum guarantee valid for any distribution and is correspondingly looser: 75% at two standard deviations rather than 95%. One is a description, the other a guarantee.
Why does Chebyshev not work for k less than 1?
Because 1 − 1/k² is zero or negative there, so the bound says at least none of the data is within the interval, which is true but useless. The theorem only becomes informative once k exceeds 1, and only becomes practically interesting from about 1.5 upward.
Is the bound ever tight?
Yes, but only for specially constructed distributions designed to attain it, which concentrate probability at exactly the right points. No naturally occurring data set sits at the bound. Real data typically has far more inside the interval than the theorem requires.
When should I use Chebyshev?
When you genuinely do not know the distribution shape and need a guarantee rather than an estimate. If you have grounds to believe the data is roughly normal, the empirical rule gives a far more useful answer. Chebyshev is the fallback, not the default.
For the tighter normal-data version, see the empirical rule calculator. For standardising a value, see the z-score calculator.